Imagine a plane flying at 3000 m above sea level. The wind crashing against its wings as it comes from several directions; its enormous weight pushing it towards the ground; its engines producing the necessary push to keep it in the air… it is definitely a complex scenario if you consider all the forces acting on the plane’s fuselage.

Now, imagine you are a plane engine designer. In order to determine the right characteristics and requirements for any of your designs you need to consider and model all the potential forces both the plane and your engine will be subjected to. This exercise can lead you to really complex scenarios involving lots of forces acting in different directions in 3D. But don’t worry! Physicists also have a great ability to simplify such scenarios and reduce problems into their basic components.

Calculating the net force acting on a body which is being subjected to several different forces can help you to significantly simplify your job when modelling complex situations. This article will guide you using simple examples to illustrate its meaning and how to calculate it step by step.

## How to calculate net force

**Draw all the forces acting on the body.****Locate the coordinate system in a way that produces the minimum number of angles with the forces.****Split each force in its horizontal and vertical components using its magnitude and the sine or cosine of each angle.****Assign the correct signs to the components depending on the selected convention.****Add the horizontal and vertical components separately.****Calculate the net force magnitude and direction based on the net horizontal and vertical components.**

## What is net force

Newton’s second law of motion relates the force applied to a body, F, and its resulting acceleration, a. It is expressed mathematically through the following equation, where the body’s mass, m, is the proportionality constant:

F=ma | (1) |

Let’s do a thought experiment to try and understand what this equation means. Imagine you are at the grocery shop and want to move a cart around to put your groceries in. If the cart was initially at rest, meaning it was not moving, and you push it, it will start to roll at a certain speed, let’s say 1 m/s. In this case, the cart went from zero velocity to 1 m/s in a very short period of time.

Since there was a change in the cart’s velocity, we can say it was *accelerated*. If that change in velocity occurred, for example, in a timeframe of one second, we can say the acceleration was exactly 1 m/s2. Go ahead and read the section called “How to measure acceleration” to understand how we got this result. Now, according to equation 1, if there was an acceleration, there had to be also a force causing it, which is the one you applied when pushing the cart. The mass in the equation is, in this case, that of the cart, which is constant. This thought experiment is summarized by this image:

Now, you have a cart to put your groceries in. Perfect. But, what would happen if somebody rushes into the store, realizes there aren’t any other carts left and decides to grab your cart? Imagine you pull on it but the other person does it as well. Will the cart be accelerated by the forces acting upon it? In which direction will it finally move? Well, that depends on who is stronger, right?

Let’s say you pull the cart with the same force as before but this time towards yourself. The person wanting to steal your cart does the same thing, but he or she is only half as strong as you. That person would be pulling the cart in the opposite direction as you with a force that is half as big as yours. In this case, the cart will end up moving towards you, since you are the stronger one, right? Nevertheless, the cart will not move as fast as before. Let’s see why:

If you were able to move the cart from 0 m/s to 1 m/s in the course of one second before, you will probably reach a lower final velocity this time, since the other person is pulling in the opposite direction, thus slowing the cart down. Now, if the cart stealer’s force is half as yours, according to equation 1 the final acceleration of the cart will be half as before, so 0,5 m/s2. This means that after one second, the cart will move at 0,5 m/s. The next image will make things clear. See how the arrows representing the forces have different lengths? This is because Fyou is twice as big as Fcart stealer.

The difference in both cases —you pulling alone and you and the cart stealer pulling in opposite directions— is that the total force acting on the cart is different, and therefore the acceleration that is caused by it is also different. That total force must then be some type of sum of all of the forces acting on the cart. However, since forces are vectors that sum is of a special type: a *vectorial sum*.

Vectors are physical entities which have a magnitude and a direction. Magnitude is the intensity of the vector and expresses how big the physical variable is. When talking about forces, the magnitude is measured in units called *Newtons*, named after sir Isaac Newton, who discovered the principle expressed by equation 1. In our previous cart-pulling example, we may say you were pulling the cart with a force of 1 Newton, or 1 N. Consequently, the cart stealer pulling opposite to you exerted a force of 0,5 N on the cart.

To simplify this exercise, we can draw the cart as a single point. Physicists use this type of simplifications all the time to make calculations easier. In this case, that point will have the same mass as the entire cart, kind of as if it had collapsed into a single point of space. This is why it is called the *center of mass*.

For all our calculations, the center of mass will have the same effect as the entire cart, and it will be represented by a gray circle. We can now draw all forces acting on this point, a.k.a. our grocery cart:

You might have noticed we drew two yellow axes right in the center of the gray circle. This is called the *coordinate system*, which is a convention that helps us know what is “up” and “down” in our imaginary experiment. We have conveniently placed it with the x-axis pointing along the line formed by the two opposing force vectors. This helps us to simplify our calculations, since no angles are formed between the vectors and any of the axes of our coordinate system.

The coordinate system lets us define that any vector or any of its components pointing to the right, in the same direction as the x-axis, will be positive. In turn, any vector or component pointing to the left will be negative. This could be the other way around and the result would be the same, so no worries! In our cart-pulling experiment, the force you exert on the cart is positive, so +1 N, while the force the cart stealer exerts is negative, -0,5 N.

Since all the forces acting on the body of interest lay along a single line, we can simply add them without the need of any further calculation: +1 N + (-0,5 N) = 0,5 N. This is the magnitude of the resulting force being applied to the cart. The direction along which those 0,5 N are being exerted is to the right of the x-axis. This is because the result of our vectorial sum is positive (remember we *defined *that everything pointing to the right of the x-axis would be positive). This way, we can calculate the resulting force that the cart experiences from your pull and that of the cart stealer combined together.

Now, if this resulting force were not the result of two forces acting at the same time, but a single force being applied by some other person alone, the effect on the cart would be exactly the same: it would accelerate at 0,5 m/s2 to the right. In summary, we can have one, two or even more forces acting together on the cart, and the result could still be the same, provided their vectorial sums are the same. This happens because in every case the *net force* is equal.

Net force is defined as a single force vector that causes the same acceleration on a body as all the individual forces acting on it. This means,** the net force is the vectorial sum of these forces**. As such, it can replace the original forces without changing the physical result. This concept is very useful when you have many forces acting on a body. With some simple calculations you can turn all of those force vectors into a single one and obtain the same effect! Let ‘s see how.

## How to find net force

Let’s recap some of the steps of our previous example and try to generalize them so they apply in many other cases. The first thing we need to do to analyze any problem involving forces acting on a body is to place the origin of our coordinate system on the most convenient spot. A common place to do so is at the center of mass of the body being analyzed. Now, what direction should the y- and x-axes be pointing in?

Use the next image as an example. The origin coincides with the center of mass and we have decided to place the x-axis in the same direction as F1. This way, the other two force vectors form angles and with the x- and the negative y-axes, respectively. Actually, placing the x-axis in the same direction as F2 or F3 would be equivalent, since the remaining two forces would then form new angles with the axes of our coordinate system.

**Tip 1: select a direction for the axes of your coordinate system that produces the least possible amount of angles between the forces and them. **

Having done so, we can start breaking down the force vectors into their components, meaning their horizontal (x) and vertical (y) parts. For this, we need a convention for what is positive and negative. A very common one is that any vector or vector component pointing to the right (positive x-axis) or upwards (positive y-axis) is positive. Consequently, any vector or vector component pointing to the left (negative x-axis) or downwards (negative y-axis) will be negative.

Separating these two perpendicular directions is very helpful, since we can then add all the horizontal components together and obtain the net horizontal force. After doing the same with the vertical components, we can then express the net force acting on the body as a single vector with its own horizontal and vertical components. Let’s do this exercise force by force:

F1 does not form any angles with the axes of our coordinate system, since it is aligned with the x-axis. This indicates its magnitude acts solely horizontally, in the direction of the x-axis. Its horizontal component is therefore positive (it points to the right) with a magnitude of F1, while its vertical component is zero. This means, no portion of this vector acts in the direction of the y-axis.

On the other hand, F2 forms an angle of with the x-axis. If an angle is formed, some portion of the vector acts horizontally and some other portion acts vertically. In order to find these components, we need trigonometric functions, such as sine and cosine. If we extract the portion of the previous drawing containing only F2 it would look like this:

In this case, according to the definition of the sine of an angle, sin =F2y/F2, where F2y is the vertical component of the force. If we solve this equation for it, we obtain: F2y=F2sin . Now, since this vertical component is pointing upwards, it is positive according to our convention. On the other hand, according to the definition of the cosine, cos =F2x/F2, where F2x is the horizontal component of F2. If we solve for the component of interest, we get F2x=F2cos . Since this vector is pointing to the right, it is positive.

**Tip 2: use the trigonometric functions sine and cosine to calculate the components of the vectors that form angles with any of your coordinate system’s axes. **

Just so you keep it in mind: **vector components are also vectors**. This means they possess a magnitude and a direction. The former is a portion of the original vector’s magnitude, and it depends on the angle it forms with one of the coordinate system’s axes. Their direction is the same as that of the axis they are parallel to.

Now we have the vertical and horizontal components of both F1 and F2. Go ahead and find the components of F3 by yourself applying the same procedure we used for F2. Pay special attention to the sign convention and remember anything pointing to the left or downwards is negative. You will find a summary of all the components of our three vectors in the next table.

**Table 1: vector components summary**

Vector | Vertical component | Horizontal component |

F1 | 0 | F1 |

F2 | F2 sin | F2 cos |

F3 | -F3 cos | -F3 sin |

Now that we have all the components of the three vectors acting upon the body we can start adding them together. The vertical and horizontal components should be added separately, keeping the signs we have assigned to each of them based on our convention. To complete this example, let’s use some real numbers. Let’s assume F1=10 N, F2=10 N, F3=7 N, =45° and =60°. Now, we can complete our table and add the results together:

**Table 2: vector components values**

Vector | Vertical component | Horizontal component |

F1 | 0 | 10 N |

F2 | 10N sin 45°=7,07 N | 10 N cos 45°=7,07 N |

F3 | -7 N cos 60°=-3,5 N | 7 N sin 60°=-6,06 N |

Total | 3,57 N | 11,01 N |

The net force acting on the body is a vector with a vertical component of 3,57 N and a horizontal component of 11,01 N. Now, how do we know its total magnitude and direction? The total magnitude of the net force can be calculated using the Pythagorean theorem: since the vertical and horizontal components are perpendicular, if we join them together they will form a right triangle.

The magnitude of the net force is the hypotenuse of the right triangle, meaning:

Fnet=Fnet-x2+Fnet-y2 | (2) |

In our example, the magnitude is: Fnet=11,57 N. To determine its direction, we just need to plot the resulting vector and everything becomes a lot clearer:

The net force points to the right and upwards, since both of its components (horizontal and vertical) are positive. It must then form an angle with the x-axis, which we call . To calculate its value, we just need to calculate its tangent: tan =Fnet-y/Fnet-x. Since we now the values of these two components, we can calculate the value of the tangent of . Finally, to find the angle, we just need to calculate the inverse tangent since tan-1(tan )=. In our example: =tan-1(3,57 N/11,01 N)=18°.

You have found the value and direction of the net force acting on a body. Remember, this vector has exactly the same effect as the original three vectors acting together, and therefore you have significantly simplified this problem!

## How to measure acceleration

Acceleration is the rate of change of velocity in time. If a car goes from 0 km/h to 80 km/h in 1 minute, which equals 0,01667 hours, the acceleration in that period of time is calculated like this:

a=Final velocity-Initial velocityTime=80 km/h – 0 km/h0,01667 h=4800 km/h2 | (3) |

This result tells us that, for every hour that passes, the car will increase its velocity in 4800 km/h! In the example of the text, the grocery cart goes from 0 m/s to 1 m/s in 1 second. This means its acceleration in that period of time was:

a=1 m/s – 0 m/s1 s=1 m/s2 | (4) |

## Other helpful sources

Use this PhET interactive simulator to understand the concept of net force in a round of tug of war, a game you have probably already played! Place the different players on each side of the rope, clic *go!* and see who wins. You can also see the different force values and the resulting speed.

## Sources:

PhET interactive simulations. University of Colorado Boulder. Consulted on May 2nd, 2021 at: https://phet.colorado.edu/sims/html/forces-and-motion-basics/latest/forces-and-motion-basics_en.html